From: Vaughan Pratt <pratt@CS.Stanford.EDU>
To: categories@mta.ca
Subject: Re: Query (Q-algebras)
Date: Thu, 08 May 2003 12:05:50 -0700 [thread overview]
Message-ID: <E19DrRz-00074B-00@mailserv.mta.ca> (raw)
In-Reply-To: Message from Vaughan Pratt <pratt@CS.Stanford.EDU> of "Wed, 07 May 2003 11:55:08 PDT." <200305071855.LAA22674@coraki.Stanford.EDU>
>(So if Z is 3 then there are 27 = 3^3 "Boolean" operations in place of
>the familiar 4 = 2^2.)
There should have been a "unary" in there of course.
Another question about these Q-algebras that Oswald Wyler was asking
about: what is a necessary and sufficient condition for a complete basis
for finitary Q-algebras (the theory of Boolean algebras rather than CABAs)
having any given Z? For Z = 2 one answer (at least for the version of the
problem which only considers nonzeroary operations) is that for each of the
following properties the basis must contain a counterexample to that property.
Necessity follows because each property is preserved under composition;
sufficiency takes more work.
* selfdual (e.g. xy+yz+zx = (x+y)(y+z)(z+x))
* monotone
* affine (expressible as the XOR of its arguments, optionally plus 1)
* strict (maps the all-zeros input to zero)
* costrict (maps the all-ones input to one)
(NAND violates all five at once.) Is there a fixed number of such properties
that works for all finite cardinalities of Z, or must the number of properties
of this kind grow with Z?
Vaughan Pratt
next prev parent reply other threads:[~2003-05-08 19:05 UTC|newest]
Thread overview: 4+ messages / expand[flat|nested] mbox.gz Atom feed top
[not found] <owyler@suscom-maine.net>
2003-05-05 17:46 ` Query Oswald Wyler
2003-05-07 18:55 ` Query (Q-algebras) Vaughan Pratt
2003-05-08 19:05 ` Vaughan Pratt [this message]
2003-05-09 11:24 ` Ernie Manes
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